September 19, 2010

Jaisalmer is known as the "Golden City," after the distinctive yellow sandstone, found locally, that is the area's main construction material. The sandstone's yellow color comes from its high sulfur content.

Constructed from yellow sandstone, Jaisalmer Fort sprawls across a hilltop in the Thar desert. More than 2,000 people live within the fort.

Located on an ancient trade route in the Thar desert, near the border between India and Pakistan, the town's strategic position brought it wealth, and its merchants and government officials built elaborate mansions (havelis) out of this yellow sandstone.

The sandstone is relatively soft, so it can be carved easily into elaborate patterns and intricate latticework, evident throughout the town.

Balcony designs can be highly intricate, with both relief carvings and latticework.

Set on either side of narrow, winding lanes within Jaisalmer Fort, these mansions stand as monuments to the anonymous stone carvers who covered seemingly every square inch with exquisite patterns. One mansion features 38 balconies, each one with a different design.

Intricate circular designs are common features of the sandstone carvings in Jaisalmer.

Latticed friezes provided ventilation and privacy for women, who could peek out without anyone seeing them.

The Taj Mahal's translucent white marble has a warm glow in the early morning light. Click on photos to see more detail.

Oriented on north-south and east-west axes, the mausoleum and its associated structures were designed around principles of reflection and repetition. The tomb itself is essentially a cube with chamfered corners, to give it an octagonal cross section (see "Octagons and Squares"). The four sides are identical, each one featuring a huge vaulted archway. Four minarets frame the tomb.

The Taj Mahal's symmetrical structure is evident in this northward view of the tomb.

Reflection symmetries also abound in the decorations, made from precious and semiprecious stones inlaid on white, translucent marble (below).

Even the reflecting pools of water add to the sense of exquisite symmetry throughout the site.

At dawn, the eastern entryway to the Taj Mahal complex.

But it's also worth looking down—at the intriguing tiling patterns of the paving stones that cover the ground around the Taj Mahal.

Next to the tomb, the stones lie in a distinctive pattern of four-pointed stars (red sandstone) and diamonds (marble).

September 11, 2010

The largest sundial I have ever seen is in the Jantar Mantar in Jaipur, India. It looms over a remarkable collection of naked-eye astronomical instruments, where large scale and geometrical ingenuity make up for the absence of optical magnification. The term "jantar" means "instrument" and "mantar" may be interpreted as "formula" or "calculation."

Constructed for Maharaja Jai Singh II at his new capital and fabricated out of masonry, marble, and bronze between 1727 and 1734, the dozen or so instruments that constitute the collection were used for making remarkably precise determinations of astronomical position without the aid of telescopes.

Together, the devices could be used to measure time, predict eclipses, track star locations, ascertain declinations of planets, and determine celestial altitudes.

The largest instrument (samrat yantra) casts a shadow that tells the time of day. Its triangular gnomon, 90 feet high, is angled at 27°, Jaipur's latitude. The triangle's hypotenuse rests parallel to Earth's axis. A quadrant of a circle lies on either side of the gnomon, parallel to the plane of the equator.

The small cupola at the top of the samrat yantra was used as a platform for announcing eclipses and the arrival of the monsoon season.

Another astronomical instrument (jai prakash yantra) consists of a pair of large hemispherical bowls, which serve as a reflection of the sky above. Crossing wires stretched across each bowl hold a centered metal ring so that every point in the sky can be reflected to a corresponding point on the bowl through the ring.

The two bowls complement each other. The open spaces in one correspond to surfaces in the other. The cutouts allow observers to view positions without inadvertently blocking the light. When a shadow falls within a cutout in one bowl, an observer simply moves to the other bowl.

Bronze devices related to astrolabes allowed the measurement of time and the positions of celestial objects.

Telling time from the smaller of the two giant sundials in the Jantar Mantar.

In July, the Jantar Mantar in Jaipur was added to the UNESCO World Heritage List as an "expression of the astronomical skills and cosmological concepts of the court of a scholarly prince at the end of the Mughal period" in Indian history.

September 9, 2010

Combinations of regular octagons and squares are often a feature of Indo-Islamic design. Such tilings appear in a variety of settings, particularly grills or screens fashioned from metal or cut out of marble or sandstone.

This doorway (above) at Golconda Fort in Hyderabad is protected by a wire grill with a distinctive pattern of overlapping octagons (below), constructed from squares and elongated hexagons.

A variant (below) of this design, with stretched octagons and rectangles, can be seen at the Red Fort in Agra.

Octagons and squares are also an important feature of the Taj Mahal and associated structures.

A structure near the Taj Mahal in Agra features an alternative combination of octagons and squares (above). Note that the interior of each octagon consists of four squares and eight pentagons.

The Taj Mahal itself (above) is essentially a square with cut-off corners to create an octagonal cross section with alternating long and short sides. The inner tomb chamber is a regular octagon.

Two overlapping squares create an eight-pointed star.

The same eight-pointed star can be seen in the Taj Mahal's gardens, surrounding by a pathway tiled with overlapping octagons made from squares and hexagons.

Amer Fort in Jaipur has a pool in the shape of an eight-pointed star, surrounding an octagonal island.

A complex octagonal design (above) carved out of red sandstone at Fatehpur Sikri.

A floor tiling combines regular octagons with octagons having alternating long and short sides.

In North America, we are used to seeing octagons in the guise of stop signs. Curiously, in India, stop signs are usually circular.

About

Welcome to an occasional series devoted to "cool stuff" that I encounter while browsing the world of mathematics and computer science. I'll peek at new developments in math and its applications, and I'll revisit old puzzles, famous problems, and historic events—anything mathematical that happens to catch my eye. I hope you'll find something of value in these brief, informal forays into the world of math.

About Me

Ivars Peterson is a freelance writer and editor. He was Director of Publications at the Mathematical Association of America from 2007 to 2014. As an award-winning mathematics writer, he previously worked at Science News for more than 25 years and served as editor of Science News Online and Science News for Kids. His books include The Mathematical Tourist, Islands of Truth, Newton's Clock, and Fragments of Infinity: A Kaleidoscope of Math and Art.